Question

An exam has 8 multiple choice questions. Each question has 4 possible answers, only one of which is correct. If a student takes this exam and answers all questions at random, what is the probability that the student answers

(a) only the first and the last question correctly?

(b) only 2 questions correctly?

(c) at least 2 questions correctly?

Answer 1

Based on the information provided in question, we can conclude that it follows binomial distribution [n = 8, p = 0. 25, q = 0. 75].

where n = 8 since there are 8 questions or trials; with a success rate of p = 0. 25 [only 1choice out of 4 choices would be the correct answer];

with a failure rate of q =0. 75 [3 choices out of 4 choices are incorrect answers].  

(a)-

Here we want to find out the probability that he attempt only the first and the last question correctly

Which will be equal to = 0.25× (0.75 )⁸ × 0.25

Because first and last answer are correct and remaining 8 answers are wrong .

Required probability = 0.0062570572

(b)- probability that he correct only 2 questions i.e.

P (X = 2 ) = (⁸ C ₂) × (0. 25) ² × (0. 75) ⁶

​​​​​P (X =2) = 0.311462402

(C) similarly

P( X2)= p(x=2) +p(x=3)+........p(x= 8)

= 1- p(x= 1)-p(x=0)

=1- C(8,1) 0.25 ¹ 0.75⁷-C(8,0) 0.75⁸

= 0.63291931152

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